Metamath Proof Explorer


Theorem cdlemeg46fgN

Description: TODO FIX COMMENT p. 116 penultimate line: f(g(r)) = r. (Contributed by NM, 4-Apr-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemef46g.b B = Base K
cdlemef46g.l ˙ = K
cdlemef46g.j ˙ = join K
cdlemef46g.m ˙ = meet K
cdlemef46g.a A = Atoms K
cdlemef46g.h H = LHyp K
cdlemef46g.u U = P ˙ Q ˙ W
cdlemef46g.d D = t ˙ U ˙ Q ˙ P ˙ t ˙ W
cdlemefs46g.e E = P ˙ Q ˙ D ˙ s ˙ t ˙ W
cdlemef46g.f F = x B if P Q ¬ x ˙ W ι z B | s A ¬ s ˙ W s ˙ x ˙ W = x z = if s ˙ P ˙ Q ι y B | t A ¬ t ˙ W ¬ t ˙ P ˙ Q y = E s / t D ˙ x ˙ W x
cdlemef46.v V = Q ˙ P ˙ W
cdlemef46.n N = v ˙ V ˙ P ˙ Q ˙ v ˙ W
cdlemefs46.o O = Q ˙ P ˙ N ˙ u ˙ v ˙ W
cdlemef46.g G = a B if Q P ¬ a ˙ W ι c B | u A ¬ u ˙ W u ˙ a ˙ W = a c = if u ˙ Q ˙ P ι b B | v A ¬ v ˙ W ¬ v ˙ Q ˙ P b = O u / v N ˙ a ˙ W a
Assertion cdlemeg46fgN K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W F G R = R

Proof

Step Hyp Ref Expression
1 cdlemef46g.b B = Base K
2 cdlemef46g.l ˙ = K
3 cdlemef46g.j ˙ = join K
4 cdlemef46g.m ˙ = meet K
5 cdlemef46g.a A = Atoms K
6 cdlemef46g.h H = LHyp K
7 cdlemef46g.u U = P ˙ Q ˙ W
8 cdlemef46g.d D = t ˙ U ˙ Q ˙ P ˙ t ˙ W
9 cdlemefs46g.e E = P ˙ Q ˙ D ˙ s ˙ t ˙ W
10 cdlemef46g.f F = x B if P Q ¬ x ˙ W ι z B | s A ¬ s ˙ W s ˙ x ˙ W = x z = if s ˙ P ˙ Q ι y B | t A ¬ t ˙ W ¬ t ˙ P ˙ Q y = E s / t D ˙ x ˙ W x
11 cdlemef46.v V = Q ˙ P ˙ W
12 cdlemef46.n N = v ˙ V ˙ P ˙ Q ˙ v ˙ W
13 cdlemefs46.o O = Q ˙ P ˙ N ˙ u ˙ v ˙ W
14 cdlemef46.g G = a B if Q P ¬ a ˙ W ι c B | u A ¬ u ˙ W u ˙ a ˙ W = a c = if u ˙ Q ˙ P ι b B | v A ¬ v ˙ W ¬ v ˙ Q ˙ P b = O u / v N ˙ a ˙ W a
15 simpl1 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W K HL W H
16 simpl3 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W Q A ¬ Q ˙ W
17 simpl2 K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W P A ¬ P ˙ W
18 simprl K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W P Q
19 18 necomd K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W Q P
20 simprr K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W R A ¬ R ˙ W
21 1 2 3 4 5 6 11 12 13 14 7 8 9 10 cdlemeg46gf K HL W H Q A ¬ Q ˙ W P A ¬ P ˙ W Q P R A ¬ R ˙ W F G R = R
22 15 16 17 19 20 21 syl32anc K HL W H P A ¬ P ˙ W Q A ¬ Q ˙ W P Q R A ¬ R ˙ W F G R = R